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The present analysis deals with the study of the $f(Q, T )$ theory of gravity which was recently considered by many cosmologists. In this theory of gravity, the action is taken as an arbitrary function $f(Q, T )$ where $Q$ is non-metricity and $T$ is the trace of energy-momentum tensor for matter fluid. In this study, we have taken three different forms of the function $f(Q, T )$ as $f(Q,T)=a_1Q+a_2 T$ and $f(Q,T)=a_3 Q^2+a_4 T$ and discussed some physical properties of the same. Also, we have obtained the various cosmological parameters towards Friedmann-Lemaitre-Robertson-walker (FLRW) Universe by defining the transit form of scale factor which yields the Hubble parameter in redshift form as $H(z)=\frac{H_{0}}{(\lambda+1)} \left(\lambda+ (1+z)^{\delta}\right)$. %We have obtained the approx best-fit values of model parameters using the least square method for observational constraints on available datasets like Hubble dataset $H(z)$, Supernova dataset SNe-Ia, etc., by applying the Root Mean Squared Error formula (RMSE). By applying the Root Mean Squared Error formula (RMSE), we were able to determine the approximate best-fit values of model parameters using the least square approach for observational constraints on the datasets Hubble dataset $H(z)$, Supernova dataset SNe-Ia, etc. We have observed that the deceleration parameter $q(z)$ exhibits a signature-flipping (transition) point within the range $0.623 \le z_{0} \le 1.668$ through which it changes its phase from the decelerated to the accelerated expanding universe with $\omega = -1$ at $z=-1$ for the approximate best-fit values of the model parameters. %For obtained approx best-fit values of model parameters we have observed that the deceleration parameter $q(z)$ shows a signature-flipping (transition) point within the range $0.623 \le z_{0} \le 1.668$ through which it changes its phase from decelerated to the accelerated expanding universe with $\omega = -1$ at $z=-1$.

Keywords:

Subject: Physical Sciences - Theoretical Physics

The most successful theory, according to the new observation, is Einstein’s theory of general relativity but it has some limitations to explain the phenomenon like the Big Bang singularity, general relativity not respecting local black holes, a consistent quantum gauge field theory of gravity, etc. The data provide compelling evidence that our universe is going through an expansion phase as well as an acceleration phase, according to [1,2,3]. For an understanding of the universe is accelerating in an organized way, from time to time various cosmologists have proposed some well-known modified theories of gravity like $f\left(G\right)$ gravity [4], $f\left(R\right)$ gravity [5,6,7], $f(R,T)$[8,9,10,11,12,13,14], $f\left(\tau \right)$ gravity [15], $f(G,T)$ gravity [16], $f(R,T,{R}_{\mu v}{T}^{\mu v})$ gravity [17], $f\left(Q\right)$ gravity [18,19,20,21,22,23,24,25] and recently proposed $f(Q,T)$ [26,27,28,29,30] gravity. Dark energy and Dark matter refer back to the unseen additives of the Universe. Dark matter is an invisible, non-baryonic matter believed to give an explanation of phenomena including gravitational lensing and galactic rotation curves. Dark energy is responsible for the accelerating expansion of the Universe [31,32].

Geometric variables in symmetric teleparallel gravity reflected the gravitational interaction’s physical characteristics, which are symbolised by the metric’s non-metricity Q. The non-metricity tensor is the covariant derivative of the metric tensor. This strategy was first presented by Nester and Yo [18]. The Lagrangian is viewed as an arbitrary function of the non-metricity in an extension of symmetric teleparallel gravity. According to Xu’s gravity theory [26], Q and the trace of the matter-energy momentum tensor T are $f(Q,T)$. As in $f(R,T)$, they cause the cosmos to undergo some large thermodynamic changes [8].

We are studying various energy conditions in the recently proposed $f(Q,T)$ gravity theory in the present work. Energy conditions play a vital role in defining cosmological evolution, the emergence of Big-Rip Singularities, and Black hole dynamics [33].It explains how geodesics behave in ways that are similar to space, time, or light. It allows us some latitude in our analysis of particular notions about the nature of cosmic geometries and particular relationships that energy momentum must satisfy under stress in order for energy density to be positive. It is typically used in general relativity to illustrate and investigate space-time singularities [34]. Using power law in $f\left(R\right)$ gravity Capozziello studied the energy condition [35]. The Null Energy Condition requires for Black hole thermodynamics [33], whereas the Hawking-Penrose singularity theorem invokes Weak Energy Condition and Strong Energy Condition [36]. M. Sharif [37] introduced energy conditions using FLRW universe for two models. The viability of the bounds in $f(R,\square \phantom{\rule{4pt}{0ex}}R,T)$ is investigated through energy conditions in [38]. In different modified theories of gravity like $f\left(R\right)$ and generalized teleparallel theory, energy conditions have been investigated [39,40,41,42].

Motivated from the above analysis and discussion in the present investigation we investigate the two different $f(Q,T)$ models in a flat FLRW space-time with proposed equation of state (EoS) $p=\omega \rho $, deceleration parameter (q) where p, $\rho $ and $\omega $ represent cosmological pressure, energy density and EoS parameter along with the validation of energy conditions.

The organisation of the analysis is as follows: The fundamental formalism of the $f(Q,T)$ theory of gravity by changing action is presented in Section II. The gravitational field equations and the emergent scale factor are shown in Section III.While the empirical constraints that explain model-free parameters are offered in Section IV. In Section V cosmological parameters are covered. In Section VI, a few models of "f(Q,T)" gravity are discussed. We have taken a function $f(Q,T)$ both linear and quadratic as $f(Q,T)={a}_{1}Q+{a}_{2}T$ and $f(Q,T)={a}_{3}{Q}^{2}+{a}_{4}T$ where ${a}_{1}$, ${a}_{2}$, ${a}_{3}$ and ${a}_{4}$ are model parameters. A summary of concluding remarks is provided in the last Section VII.

The modified Einstein-Hilbert action principle for the $f(Q,T)$ extended symmetric teleparallel gravity is given by [26]
where $f(Q,T)$ being the general functional form of the non-metricity scalar Q and the trace of the energy-momentum tensor T. g is the determinant of the metric tensor ${g}_{\mu \nu}$ i.e. $g=det\left(\right)open="("\; close=")">{g}_{\mu \nu}$, and ${L}_{m}$ is Lagrangian matter. The non-metricity scalar Q is defined as
where the deformation tensor ${L}^{\delta}{}_{\alpha \gamma}$ is given by

$$S=\int \left(\right)open="["\; close="]">\frac{1}{16\pi}f(Q,T)+{L}_{m}$$

$$Q\equiv -{g}^{\mu \nu}\left(\right)open="("\; close=")">{L}^{\delta}{}_{\alpha \mu}{L}^{\alpha}{}_{\nu \delta}-{L}^{\delta}{}_{\alpha \delta}{L}^{\alpha}{}_{\mu \nu}$$

$${L}^{\delta}{}_{\alpha \gamma}=-\frac{1}{2}{g}^{\delta \eta}\left(\right)open="("\; close=")">{\nabla}_{\gamma}{g}_{\alpha \eta}+{\nabla}_{\alpha}{g}_{\eta \gamma}-{\nabla}_{\eta}{g}_{\alpha \gamma}$$

The non-metricity tensor has the following definition:
and the non-metricity tensor’s trace is derived as

$${Q}_{\gamma \mu \nu}={\nabla}_{\gamma}{g}_{\mu \nu},$$

$${Q}_{\delta}={g}^{\mu \nu}{Q}_{\delta \mu \nu},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}{\tilde{Q}}_{\delta}={g}^{\mu \nu}{Q}_{\mu \delta \nu}.$$

Further, define the super potential tensor as follows
using this the non-metricity scalar is

$${P}_{\mu \nu}^{\delta}=-\frac{1}{2}{Q}_{\mu \nu}^{\delta}+\frac{1}{4}\left(\right)open="("\; close=")">{Q}^{\delta}-{\tilde{Q}}^{\delta}$$

$$Q=-{Q}_{\delta \mu \nu}{P}^{\delta \mu \nu}.$$

The variation of the energy-momentum tensor with respect to the metric tensor ${g}_{\mu \nu}$ read as
where the energy-momentum tensor is
and

$$\frac{\delta \left({g}^{\mu \nu}{T}_{\mu \nu}\right)}{\delta {g}^{\alpha \beta}}={T}_{\alpha \beta}+{\theta}_{\alpha \beta}.$$

$${T}_{\mu \nu}=\frac{-2}{\sqrt{-g}}\frac{\delta \left(\right)open="("\; close=")">\sqrt{-g}{L}_{m}}{}\delta {g}^{\mu \nu}$$

$${\theta}_{\mu \nu}={g}^{\alpha \beta}\frac{\delta {T}_{\alpha \beta}}{\delta {g}^{\mu \nu}}.$$

Also, the field equations of $f\left(\right)open="("\; close=")">Q,T$ gravity are given by varying the action, Equation (1), with respect to the metric tensor ${g}_{\mu \nu}$,
where ${f}_{Q}=\frac{df\left(\right)open="("\; close=")">Q,T}{}dQ$, ${f}_{T}=\frac{df\left(\right)open="("\; close=")">Q,T}{}dT$, and ${\nabla}_{\delta}$ denotes the covariant derivative. From Equation (11) it emerge that the field equations of $f\left(\right)open="("\; close=")">Q,T$ depend on the tensor ${\theta}_{\mu \nu}$. Various cosmological models of $f\left(\right)open="("\; close=")">Q,T$ gravity, depending on the nature of the source of matter are possible. Very recently Koussour et al. [43] have investigated the quintessence form of extended symmetric teleparallel gravity with cosmic acceleration by assuming the cosmic time-redshift relation as $t\left(z\right)=\frac{n{t}_{0}}{m}g\left(z\right)$ which gives the Hubble parameter and verify the sustainability of the results through the energy conditions along with jerk parameter. Also, in [44] the authors investigated energy conditions in $f(Q,T)$ gravity for two different forms of models.

$$-\frac{2}{\sqrt{-g}}{\nabla}_{\delta}\left(\right)open="("\; close=")">{f}_{Q}\sqrt{-g}{P}^{\delta}{}_{\mu \nu}-{f}_{Q}\left(\right)open="("\; close=")">{P}_{\mu \delta \alpha}{Q}_{\nu}{}^{\delta \alpha}-2{Q}^{\delta \alpha}{}_{\mu}{P}_{\delta \alpha \nu}$$

In order to find a solution for the field equations in $f(Q,T)$ extended symmetric teleparallel gravity, some straightforward assumptions, such as the selection of a metric, are frequently required. Consequently, we consider the form’s flat FLRW metric,
where $a\left(t\right)$ is the scale factor of the metric and its depends on cosmic time (where unit of the cosmic time is Gyr). Energy-momentum tensor for the Universe, imagine to be behaves like a perfect fluid is given by

$$d{s}^{2}=-d{t}^{2}+{a}^{2}\left(t\right)\left(\right)open="("\; close=")">d{x}^{2}+d{y}^{2}+d{z}^{2}$$

$${T}_{\nu}^{\mu}=diag\left(\right)open="("\; close=")">-\rho ,p,p,p$$

Here, p denotes pressure and $\rho $ denotes energy density of the Universe. Thus, for tensor ${\theta}_{\nu}^{\mu}$, the expression is obtained as ${\theta}_{\nu}^{\mu}=diag\left(\right)open="("\; close=")">2\rho +p,-p,-p,-p$. The Einstein field equations using the metric (12) are given as,

$${\kappa}^{2}\rho =\frac{f}{2}-6F{H}^{2}-\frac{2\tilde{G}}{1+\tilde{G}}\left(\right)open="("\; close=")">\stackrel{.}{F}H+F\stackrel{.}{H}$$

$${\kappa}^{2}p=-\frac{f}{2}+6F{H}^{2}+2\left(\right)open="("\; close=")">\stackrel{.}{F}H+F\stackrel{.}{H}$$

Hence overhead $\left(\text{\xb7}\right)$ represents a derivative with respect to cosmic time $\left(t\right)$. In this case, $F\equiv {f}_{Q}$ and ${\kappa}^{2}\tilde{G}\equiv {f}_{T}$ represent differentiation of function $f(Q,T)$ with respect to Q and T respectively and $Q=6{H}^{2}$ . With the help of Equations (14) and (15), the EoS parameter is expressed as

$$\omega =-1+\left(\right)open="("\; close=")">\frac{1}{{\kappa}^{2}\rho}\left(\right)open="("\; close=")">\stackrel{.}{F}H+F\stackrel{.}{H}$$

Now we provide a glimpse on the main features of the scale factor (transit scale factor) and derive a few physical quantities from them to discuss the observed scenario. Hence for the transit scale factor, Hubble’s parameter is observed as
where $\u03f5$, $\lambda $ and $\delta $ are the model parameters. Keep in mind the relation of a and z as $a=\frac{1}{1+z}$, Equation (17) becomes
here ${H}_{0}$ represents the present Hubble constant which explains the present expansion rate of the Universe. Freedman et al. [45] and Suyu et al. [46] evaluated a value of the present Hubble constant ${H}_{0}=72\pm 8km/s/Mpc$ and $69.{7}_{-5}^{+4.9}km/s/Mpc$ where as recently Plank gives ${H}_{0}=67.3\pm 1.20km/s/Mpc$ [47].

$$H\left(z\right)=\u03f5\left(\right)open="("\; close=")">{a}^{-\delta}+\lambda $$

$$H\left(z\right)=\frac{{H}_{0}}{(\lambda +1)}\left(\right)open="("\; close=")">\lambda +{(1+z)}^{\delta}$$

Using the time-redshift differential relation $\frac{d}{dt}=-(1+z)H\frac{d}{dz}$, the first and second derivative of H is obtained as

$$\begin{array}{c}\hfill \dot{H}\left(z\right)=-\frac{\delta {H}_{0}^{2}{(z+1)}^{\delta}\left(\right)open="("\; close=")">\lambda +{(z+1)}^{\delta}}{}{(\lambda +1)}^{2}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\end{array}$$

It is interesting to observe and compare our investigated model with the recent observations by finding the best-fit values and the best-fit curve of the Hubble function towards the model parameters ${H}_{0}$, $\u03f5$, $\lambda $, and $\delta $ with the recent observational datasets. Hence, here we used the most favorable Hubble and Supernovae SNe-Ia to constrain the said model parameter in the following section.

The accelerated expansion can be explained by a cosmological constant; alternative explanations, such as dynamical dark energy or modified gravity, can be investigated by looking at how they affect the history of the universe’s late-time expansion or the development of its structures. In order to identify constant parameters, we used the Hubble and Supernova Ia Datasets in this research.

We notice the observational constraints on the parameters ${H}_{0}=\u03f5(\lambda +1)$, $\delta $ and $\lambda $ using the latest 54 data points of $H\left(z\right)$ in the red-shift range $0.07\le z\le 2.4$ in which 28 points obtained using DA method, whereas 26 points using BAO. The values are presented in Table 2.

We have found best-fit curve of $H\left(z\right)$ with 54 data observed values shown in Table 2, using ${R}^{2}-test$:

$${R}^{2}=1-\frac{{\sum}_{1}^{54}{[{\left({H}_{i}\right)}_{obs}-{\left({H}_{i}\right)}_{th}]}^{2}}{{\sum}_{1}^{54}{[{\left({H}_{i}\right)}_{obs}-{\left({H}_{i}\right)}_{mean}]}^{2}}$$

If ${R}^{2}=1$ shows an exact fit the values of model parameters ${H}_{0}$,$\delta $ and $\lambda $ with observational datasets. From Equation (17) we required $-1<z$ and $\u03f5\ne 0$. To find the best-fit values of ${H}_{0}$ ,$\delta $ and $\lambda $ we have restricted the parametric space $-1<z$ and $\u03f5\ne 0$. We used error bars to represent the mean point and its deviation from the mean for 54 points of Hubble data-set and compared our model with the well-accepted $\Lambda $ CDM model for ${H}_{0}=67.8km/s/Mpc$, ${\Omega}_{{\Lambda}_{0}}=0.7$ and ${\Omega}_{{m}_{0}}=0.3$. We have obtained the best-fit plot for approx values ${H}_{0}=64.{49}_{-0.32}^{+0.33}$,$\delta =1.{54}_{-0.02}^{+0.02}$ and $\lambda =1.{14}_{-0.077}^{0.068}$ having maximum ${R}^{2}=0.9321$ with RMSE is 11.071 as shows in Figure 1. Therefore ${H}_{0}=64.{4772}_{-0.32}^{+0.33}km/s/Mpc$ with $6.79\%$ away from exact fit. The Figure 3 show the $1-\sigma $ (dark blue shaded), $2-\sigma $ (Sky blue shaded) maximum likelihood contours in the ${H}_{0}$-$\delta $, ${H}_{0}$-$\lambda $ and $\lambda $-$\delta $ planes for Hubble Datasets.

Distance modulus $\mu \left(z\right)$=$m-M$ given by
where M is constant for all SNe Ia. To obtain the best-fit curve we have to consider data frame having 580 entries observed of apparent magnitude from union 2.1 compilation [48] where ${d}_{L}\left(z\right)=(1+z){\int}_{0}^{z}\frac{{H}_{0}}{H\left({z}^{*}\right)}d{z}^{*}$. From the [48] statistically significant value of M is -19.30.

$$\mu \left(z\right)=5{log}_{10}{d}_{L}-5{log}_{10}\left(\frac{{H}_{0}}{Mpc}\right)+25$$

$${R}^{2}=1-\frac{{\sum}_{1}^{580}{[{\left({\mu}_{i}\right)}_{obs}-{\left({\mu}_{i}\right)}_{th}]}^{2}}{{\sum}_{1}^{580}{[{\left({\mu}_{i}\right)}_{obs}-{\left({\mu}_{i}\right)}_{mean}]}^{2}}$$

If ${R}^{2}=1$ shows exact fit the values of model parameters ${H}_{0}$ ,$\delta $ and $\lambda $ with observational datasets. We used error bars to represent the mean point and its deviation from the mean for 580 points of SNe-Ia Dataset and compared our model with the well-accepted $\Lambda $ CDM model for ${H}_{0}=67.8km/s/Mpc$, ${\Omega}_{{\Lambda}_{0}}=0.7$ and ${\Omega}_{{m}_{0}}=0.3$. We have obtained the best-fit plot for approx values ${H}_{0}=68.{665}_{-2.1}^{+2.2}$,$\delta =1.{53}_{-0.29}^{+0.28}$ and $\lambda =1.{86}_{-0.34}^{+0.37}$ having maximum ${R}^{2}=0.9930$ with RMSE is 0.2662 as shows in Figure 2.

Therefore ${H}_{0}=68.{665}_{-2.1}^{+2.2}km/s/mpc$ with $0.7\%$ away from exact fit. The Figure 3 show the $1-\sigma $ (dark green shaded), $2-\sigma $ (Light green shaded) maximum likelihood contours in the ${H}_{0}$-$\delta $, ${H}_{0}$-$\lambda $ and $\lambda $-$\delta $ planes for SNe-Ia Datasets.

Immediately, we accepted a set of solutions for the planned plan. In order to talk about how Universe is evolved in various phases, we need discuss the behavior of some cosmological parameters like the deceleration parameter, statefinder parameter etc., and these are defined as:

**The deceleration parameter $\left(\mathit{q}\right)$ is**

$$q=-1+\frac{d}{dt}\left(\right)open="("\; close=")">\frac{1}{H}$$

Here the decelerating phase refers to $q>0$ while $q<0$ corresponds to the accelerating phase of the universe. for $q=0$ is the transition point for the Universe from the deceleration to the acceleration phase.

**The statefinder parameters are**

$$r\left(z\right)=1+3\frac{\dot{H}}{{H}^{2}}+\frac{\ddot{H}}{{H}^{3}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}s\left(z\right)=q+2{q}^{2}-\frac{\dot{q}}{H}.$$

The flat $\Lambda $CDM model is shown at the point where the statefinder parameters $\left(\right)open="\{"\; close="\}">r,\mathrm{s}$ have the corresponding values. Additionally, keep in mind that in the $\left(\right)$ plane, a positive parameter s (i.e., $s>0$) denotes a quintessence-like model of dark energy, whereas a negative parameter s (i.e., $s<0$) denotes a phantom-like model of dark energy. Furthermore, by traversing the point $\left(\right)open="\{"\; close="\}">r,\mathrm{s}$, one can evolve from a phantom to the quintessence.

**Energy conditions**

One can derive highly potent and broad theorems regarding the behavior of massive gravitational fields and cosmic geometries using the energy conditions (ECs) of general relativity (GR). Generally speaking, ECs can be divided into

- Strong Energy Condition (SEC): Gravity should always be attractive and it formulated as $\rho +3p\ge 0$.
- Dominant Energy Condition (DEC): When observer measures a matter energy density then will be positive and propagate in a causal way, which leads to $\rho \ge \left|p\right|$.
- Weak Energy Condition (WEC): The matter energy density measured by any observer should be positive, $\rho \ge 0,\rho +p\ge 0$.
- Null Energy Condition (NEC): It’s the minimum requirement that is implied by SEC and WEC, is $\rho +p\ge 0$.

The violation of the NEC in the energy conditions implies that none of the energy criteria given are valid. The current fast expansion of the universe has raised questions about the SEC. In cosmological situations during the inflationary expansion and at the present, SEC must be broken.

Making use of Equations (18) and (19), the deceleration parameter and statefinder parameters are obtained as

$$q=-1+\frac{\delta {(z+1)}^{\delta}}{\lambda +{(z+1)}^{\delta}}.$$

$$r\left(z\right)=\frac{{\lambda}^{2}+(\delta -2)(\delta -1)\lambda {(z+1)}^{\delta}+(\delta -1)(2\delta -1){(z+1)}^{2\delta}}{{\left(\right)}^{\lambda}}$$

$$s\left(z\right)=1+\frac{\delta {(z+1)}^{\delta -1}\left(\right)open="("\; close=")">(z+1)\u03f5\left(\right)open="("\; close=")">(2\delta -3){(z+1)}^{\delta}-3\lambda}{\left(\right)}-\delta \lambda \u03f5{\left(\right)}^{\lambda}3$$

After analysing the SNe-Ia data by many researchers it was observed that datasets favour current acceleration for $(z<0.5)$ and past deceleration for $(z>0.5)$. A little while back, according to the high-z supernova search (HZSNS) team ${z}_{0}=0.46\pm 0.130$ at (1$\sigma $ ) confidence level [49] which has been further analyzed to ${z}_{0}=0.43\pm 0.070$ at (1 $\sigma $ ) [49]. According to SNLS [50], as well as the one recently compiled in [51], the transition red-shift ${z}_{0}$≡ 0.6 (1$\sigma $) is in better agreement with the flat $\Lambda $CDM model ${z}_{0}={(2{\Omega}_{\Lambda}/{\Omega}_{m})}^{1/3}-1\sim 0.66)$. Another limit is $0.60\le {z}_{0}\le 1.18$ (2$\sigma $, joint analysis) [52]. Further, the transition red-shift for our derived model comes to be ${z}_{0}\cong 0.65$ for observed Hubble datasets and ${z}_{0}\cong 1.965$ for supernovae which is in best agreement with the SNe- Ia supernovae observations, including the farthest known supernova SNI997ff at $z\approx 1.7$ [53]. We see that the variation of q with z obtained in our model is compatible with the results. In our derived model the best-fit value of deceleration parameter ${q}_{0}$ for Hubble and Supernova is -0.2792 , -0.4774. Figure 4 shows plot of deceleration parameter v Redshift for both Hubble and Supernovae datasets for values of model parameters ${H}_{0}$, $\delta $ and $\lambda $ are from Table 1.

In this study, it was argued that $\left(\right)$ plane is useful to differentiate between various models. An analysis based on $\left(\right)$ has also useful to differentiate between general relativity and modified theory of gravity. We note that for the Hubble datasets, the r and s parameters at the present epoch are ${r}_{0}=0.43987$ and ${s}_{0}=-3.3632$ while for the SNe-Ia datasets, ${r}_{0}=0.48918$ and ${s}_{0}=-10.5851$ as shown in Figure 5. Currently, observations are not sensitive enough to measure these parameters. Future data, however, could be used to infer these values, which would tremendously aid in defining the characteristics of dark energy.

In this section, we discuss some physical aspects of different models of $f(Q,T)$ gravity

Here we consider the $f(Q,T)$ gravity model as

$$f(Q,T)={a}_{1}Q+{a}_{2}T,$$

For above model the field Equations (14)–(16), take the form

$$\rho =\frac{{a}_{1}\dot{H}}{{a}_{2}+8\pi}-\frac{{a}_{1}\left(\right)open="("\; close=")">3{H}^{2}+\dot{H}}{}2\left(\right)open="("\; close=")">{a}_{2}+4\pi ,$$

Figure 6 shows a plot of Pressure versus Redshift for both Hubble and SNe-Ia datasets for ${H}_{0}$, $\delta $ and $\lambda $ are from Table 1 while model parameters ${a}_{1}$=-0.0125 and ${a}_{2}$=-0.012 respectively.

$$\omega =\frac{-3{a}_{2}\left(\right)open="("\; close=")">{H}^{2}+\dot{H}}{-}$$

The EoS parameter is associated with energy density $\rho $ and pressure p. The EoS parameter appears to be positive in the beginning. As a consequence, it moves from the positive region to negative region. The negative $\omega $ is proposed a constant vacuum energy density, It’s worth noting that $\omega =0$ shows Pressure-less Cold matter (PCL), $\omega =(0,\frac{1}{3})$ represents Hot matter, $\omega =\frac{1}{3}$ is radiation, $\omega =(\frac{1}{3},1)$ is Hard Universe, $\omega =1$ shows stiff fluid (SF), $\omega >1$ is Ekpyrotic matter (Ek-M), $\omega >-1$ stand for the quintessence (Q) region and $\omega <-1$ stands for the phantom region (Ph) , respectively while $\omega =-1$ represents the cosmological constant ($\Lambda $ CDM) and $\omega <<-1$ is precluded by SNe-Ia perceptions. Subsequently, the evolving range of $\omega $ of our derived model is supportive of ($\Lambda $ CDM) model in both Hubble and supernova data.

From Figure 7, we can observe that the Universe exists the decelerated regime and enters in the accelerating phase as studied [54].

Furthermore, to verify the genuineness of model in context of cosmic acceleration, we resolve different forms of energy conditions by calculating
and
for NEC, DEC and SEC respectively. Figure 8 and Figure 9 are plots of Energy Conditions with respect to constant obtained from best-fit for Hubble and SNe-Ia datasets as shown in Table 1 and model parameters ${a}_{1}$=-0.0125 and ${a}_{2}$=-0.012. According to the both data of the accelerating Universe, the SEC must be violated on cosmological scale [55,56]. Also, negative EoS ( $\omega $) indicate that $\rho +3p<0$. Therefore, there is a violation of the SEC at present. We also can see in Figure 8 and Figure 9 that the NEC, and DEC are obeying for both Hubble and SNe-Ia datasets. Since we have shown the behavior of energy density in Figure 6. We have examined the NEC behavior (i.e., partial condition of WEC). Therefore, validation of NEC and energy density together results in the validation of WEC.

$$\rho +p=\frac{2{a}_{1}\dot{H}}{{a}_{2}+8\pi},$$

$$\rho -p=-\frac{{a}_{1}\left(\right)open="("\; close=")">3{H}^{2}+\dot{H}}{}{a}_{2}+4\pi $$

$$3p+\rho ={a}_{1}\left(\right)open="("\; close=")">\frac{3{H}^{2}+\dot{H}}{{a}_{2}+4\pi}+\frac{4\dot{H}}{{a}_{2}+8\pi}$$

z | H(z) | ${\mathit{\sigma}}_{\mathit{H}}$ | Ref. | z | H(z) | ${\mathit{\sigma}}_{\mathit{H}}$ | Ref. |
---|---|---|---|---|---|---|---|

0.07 | 69 | 19.6 | [57] | 0.9 | 69 | 12 | [58] |

0.120 | 68.6 | 26.2 | [57] | 0.170 | 83 | 8 | [59] |

0.179 | 75 | 4 | [60] | 0.2 | 72.9 | 29.6 | [57] |

0.27 | 77 | 14 | [59] | 0.28 | 88.8 | 36.6 | [57] |

0.350 | 76.3 | 5.6 | [61] | 0.38 | 83 | 13.5 | [62] |

0.4 | 95 | 17 | [59] | 0.42 | 87.1 | 11.2 | [62] |

0.44 | 92.8 | 12.9 | [62] | 0.47 | 89 | 34 | [57] |

0.48 | 97 | 62 | [63] | 0.6 | 87.9 | 6.1 | [64] |

0.68 | 92 | 8 | [60] | 0.73 | 97.3 | 7 | [64] |

0.78 | 105 | 12 | [60] | 0.87 | 125 | 17 | [60] |

0.90 | 117 | 23 | [59] | 1.037 | 154 | 20 | [60] |

1.3 | 168 | 17 | [59] | 1.363 | 160 | 33.6 | [59] |

1.430 | 177 | 18 | [59] | 1.530 | 140 | 14 | [59] |

1.750 | 202 | 40 | [59] | 1.965 | 186.5 | 50.4 | [65] |

0.24 | 79.69 | 2.99 | [66] | 0.30 | 81.7 | 6.22 | [67] |

0.31 | 78.18 | 4.74 | [68] | 0.34 | 83.8 | 3.66 | [66] |

0.35 | 87.7 | 9.1 | [69] | 0.36 | 79.94 | 3.38 | [68] |

0.38 | 81.5 | 1.9 | [70] | 0.40 | 82.04 | 2.03 | [68] |

0.43 | 86.45 | 3.97 | [66] | 0.44 | 82.6 | 7.8 | [71] |

0.44 | 84.81 | 1.83 | [68] | 0.48 | 87.79 | 2.03 | [68] |

0.51 | 90.4 | 1.9 | [70] | 0.52 | 94.35 | 2.64 | [68] |

0.56 | 93.34 | 2.3 | [68] | 0.57 | 87.6 | 7.8 | [72] |

0.57 | 96.8 | 3.4 | [73] | 0.59 | 98.48 | 3.18 | [68] |

0.6 | 87.9 | 6.1 | [71] | 0.61 | 97.3 | 2.1 | [70] |

0.64 | 98.82 | 2.98 | [68] | 0.73 | 97.3 | 7 | [71] |

2.30 | 224 | 8.6 | [74] | 2.33 | 224 | 8 | [75] |

2.34 | 222 | 8.5 | [76] | 2.36 | 226 | 9.3 | [77] |

Here we consider the $f(Q,T)$ gravity model as,

$$f(Q,T)={a}_{3}{Q}^{2}+{a}_{4}T,$$

For above model the field Equations (14)–(16), take the form

$$\rho =-\frac{3\left(\right)open="("\; close=")">72\pi {a}_{3}{H}^{4}+9{a}_{3}{a}_{4}{H}^{4}-2{a}_{3}{a}_{4}{H}^{2}\dot{H}-4{a}_{3}{a}_{4}\dot{H}{H}^{2}}{}\left(\right)open="("\; close=")">{a}_{4}+4\pi \left(\right)open="("\; close=")">{a}_{4}+8\pi $$

Figure 10 shows plot of Density versus redshift for both Hubble and SNe-Ia datasets for ${H}_{0}$, $\delta $ and $\lambda $ are from Table 1, while model parameters ${a}_{1}$=-0.0125 and ${a}_{2}$=-0.012 respectively.

$$\omega =\frac{-3{a}_{4}\left(\right)open="("\; close=")">3{H}^{2}+4\dot{H}+2\dot{H}}{-}$$

As shown in Figure 11, the universe is in an accelerating mode and is about to enter a decelerating phase. Additionally, we resolve various energy conditions by calculating the model’s accuracy in the context of cosmic acceleration.
and
for NEC, DEC and SEC, respectively.

$$\rho +p=\frac{24{a}_{3}{H}^{2}\left(\right)open="("\; close=")">2\dot{H}+\dot{H}}{}{a}_{4}+8\pi $$

$$\rho -p=-\frac{6{a}_{3}{H}^{2}\left(\right)open="("\; close=")">9{H}^{2}+4\dot{H}+2\dot{H}}{}{a}_{4}+4\pi $$

$$3p+\rho =\frac{6{a}_{3}{H}^{2}\left(\right)open="("\; close=")">{a}_{4}\left(\right)open="("\; close=")">9{H}^{2}+20\dot{H}+10\dot{H}}{+}\left(\right)open="("\; close=")">{a}_{4}+4\pi \left(\right)open="("\; close=")">{a}_{4}+8\pi $$

For NEC, DEC and SEC respectively. Figure 12 and Figure 13 are plots of Energy Conditions with respect to constant obtained from best-fit for Hubble and SNe-Ia datasets as shown in Table 1 and model parameters ${a}_{1}$=-0.0125 and ${a}_{2}$=-0.012. According to the both data of the accelerating Universe,as shown in Figure 12 and Figure 13 the SEC must be violated on cosmological scale [55,56]. Also, negative EoS ( $\omega $) indicate that $\rho +3p<0$. Therefore, there is a violation of the SEC at present. Figure 12 and 13 indicate that the NEC, and DEC are obeying for both Hubble and SNe-Ia datasets. We have examined the NEC behavior (i.e., partial condition of WEC). Therefore, validation of NEC and energy density together results in the validation of WEC.

In present work we have taken a function $f(Q,T)$ both linear and quadratic as
where ${a}_{1}$, ${a}_{2}$, ${a}_{3}$ and ${a}_{4}$ are model parameters. In terms of redshift z, we have measured a number of cosmological parameters in the FLRW universe, including the Hubble parameter H and the deceleration parameter q. By applying the ${R}^{2}-test$ formula for observational constraints on the model, we were able to determine the approximate best-fit values of the model parameters $\u03f5,\delta ,\lambda ,\mathrm{and}{H}_{0}$ utilizing datasets like the Hubble data set $H\left(z\right)$ and union 2.1 compilation of SNe-Ia datasets. The current values of the cosmological models ${H}_{0}$ and ${q}_{0}$ that we estimated are close to the values found in mainstream cosmology. Talk about EoS and the various energy conditions. Following are the characteristics of our cosmological model:

$$f(Q,T)={a}_{1}Q+{a}_{2}T,f(Q,T)={a}_{3}{Q}^{2}+{a}_{4}T$$

- Figure 1 and Figure 2 provide best-fit plots based on the observed datasets. For the best fit, we employed a hybrid model combining the gradient descent and least squares approach. ${R}^{2}$-value for Hubble and SNe-Ia datasets are $0.9321$ and $0.9930$ respectively. SNe-Ia has 580 observation and giving best-fit among both datasets.
- Hubble function derived is constrained by observational datasets i.e. Hubble and SNe-Ia datasets and the present value of Hubble constant is $64.4772km/s/Mpc$ and $68.665km/s/Mpc$ respectively with respect to best-fit plot.
- The transition from early deceleration to the universe’s present acceleration is shown by the deceleration parameter ${q}_{0}=-0.2792$ and ${q}_{0}=-0.4774$ with respect to Hubble and SNe-Ia datasets.
- We have considered two functional form of $f(Q,T)$ gravity in Section A, Section B and watched how the energy density and EoS parameter all behaved. The EoS parameter exhibits a transition from early deceleration to late time acceleration with regard to the model parameters, while the density in both models exhibits positive behavior.

No advanced data associated with this article.

The IUCAA, Pune, India, provided the facilities and support under the Visiting Associateship Programmes, for which the authors (S. H. Shekh and A. Pradhan) are grateful.

The authors declared that they have neither personal relationships nor competing financial interests that could influenced the work reported in this paper.

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Datasets | ${\mathit{H}}_{0}$ | $\mathit{\delta}$ | $\mathit{\lambda}$ | $\mathit{\u03f5}$ |
---|---|---|---|---|

Hubble | $64.{49}_{-0.32}^{+0.33}$ | $1.{54}_{-0.02}^{+0.02}$ | $1.{14}_{-0.077}^{+0.068}$ | $30.{2}_{-0.87}^{+0.90}$ |

SNe-Ia | $68.{665}_{-2.1}^{+2.2}$ | $1.{53}_{-0.29}^{+0.28}$ | $1.{86}_{-0.34}^{+0.37}$ | $23.{954}_{-2.84}^{+3.74}$ |

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The present analysis deals with the study of the $f(Q, T )$ theory of gravity which was recently considered by many cosmologists. In this theory of gravity, the action is taken as an arbitrary function $f(Q, T )$ where $Q$ is non-metricity and $T$ is the trace of energy-momentum tensor for matter fluid. In this study, we have taken three different forms of the function $f(Q, T )$ as $f(Q,T)=a_1Q+a_2 T$ and $f(Q,T)=a_3 Q^2+a_4 T$ and discussed some physical properties of the same. Also, we have obtained the various cosmological parameters towards Friedmann-Lemaitre-Robertson-walker (FLRW) Universe by defining the transit form of scale factor which yields the Hubble parameter in redshift form as $H(z)=\frac{H_{0}}{(\lambda+1)} \left(\lambda+ (1+z)^{\delta}\right)$. %We have obtained the approx best-fit values of model parameters using the least square method for observational constraints on available datasets like Hubble dataset $H(z)$, Supernova dataset SNe-Ia, etc., by applying the Root Mean Squared Error formula (RMSE). By applying the Root Mean Squared Error formula (RMSE), we were able to determine the approximate best-fit values of model parameters using the least square approach for observational constraints on the datasets Hubble dataset $H(z)$, Supernova dataset SNe-Ia, etc. We have observed that the deceleration parameter $q(z)$ exhibits a signature-flipping (transition) point within the range $0.623 \le z_{0} \le 1.668$ through which it changes its phase from the decelerated to the accelerated expanding universe with $\omega = -1$ at $z=-1$ for the approximate best-fit values of the model parameters. %For obtained approx best-fit values of model parameters we have observed that the deceleration parameter $q(z)$ shows a signature-flipping (transition) point within the range $0.623 \le z_{0} \le 1.668$ through which it changes its phase from decelerated to the accelerated expanding universe with $\omega = -1$ at $z=-1$.

Keywords:

Subject: Physical Sciences - Theoretical Physics

The most successful theory, according to the new observation, is Einstein’s theory of general relativity but it has some limitations to explain the phenomenon like the Big Bang singularity, general relativity not respecting local black holes, a consistent quantum gauge field theory of gravity, etc. The data provide compelling evidence that our universe is going through an expansion phase as well as an acceleration phase, according to [1,2,3]. For an understanding of the universe is accelerating in an organized way, from time to time various cosmologists have proposed some well-known modified theories of gravity like $f\left(G\right)$ gravity [4], $f\left(R\right)$ gravity [5,6,7], $f(R,T)$[8,9,10,11,12,13,14], $f\left(\tau \right)$ gravity [15], $f(G,T)$ gravity [16], $f(R,T,{R}_{\mu v}{T}^{\mu v})$ gravity [17], $f\left(Q\right)$ gravity [18,19,20,21,22,23,24,25] and recently proposed $f(Q,T)$ [26,27,28,29,30] gravity. Dark energy and Dark matter refer back to the unseen additives of the Universe. Dark matter is an invisible, non-baryonic matter believed to give an explanation of phenomena including gravitational lensing and galactic rotation curves. Dark energy is responsible for the accelerating expansion of the Universe [31,32].

Geometric variables in symmetric teleparallel gravity reflected the gravitational interaction’s physical characteristics, which are symbolised by the metric’s non-metricity Q. The non-metricity tensor is the covariant derivative of the metric tensor. This strategy was first presented by Nester and Yo [18]. The Lagrangian is viewed as an arbitrary function of the non-metricity in an extension of symmetric teleparallel gravity. According to Xu’s gravity theory [26], Q and the trace of the matter-energy momentum tensor T are $f(Q,T)$. As in $f(R,T)$, they cause the cosmos to undergo some large thermodynamic changes [8].

We are studying various energy conditions in the recently proposed $f(Q,T)$ gravity theory in the present work. Energy conditions play a vital role in defining cosmological evolution, the emergence of Big-Rip Singularities, and Black hole dynamics [33].It explains how geodesics behave in ways that are similar to space, time, or light. It allows us some latitude in our analysis of particular notions about the nature of cosmic geometries and particular relationships that energy momentum must satisfy under stress in order for energy density to be positive. It is typically used in general relativity to illustrate and investigate space-time singularities [34]. Using power law in $f\left(R\right)$ gravity Capozziello studied the energy condition [35]. The Null Energy Condition requires for Black hole thermodynamics [33], whereas the Hawking-Penrose singularity theorem invokes Weak Energy Condition and Strong Energy Condition [36]. M. Sharif [37] introduced energy conditions using FLRW universe for two models. The viability of the bounds in $f(R,\square \phantom{\rule{4pt}{0ex}}R,T)$ is investigated through energy conditions in [38]. In different modified theories of gravity like $f\left(R\right)$ and generalized teleparallel theory, energy conditions have been investigated [39,40,41,42].

Motivated from the above analysis and discussion in the present investigation we investigate the two different $f(Q,T)$ models in a flat FLRW space-time with proposed equation of state (EoS) $p=\omega \rho $, deceleration parameter (q) where p, $\rho $ and $\omega $ represent cosmological pressure, energy density and EoS parameter along with the validation of energy conditions.

The organisation of the analysis is as follows: The fundamental formalism of the $f(Q,T)$ theory of gravity by changing action is presented in Section II. The gravitational field equations and the emergent scale factor are shown in Section III.While the empirical constraints that explain model-free parameters are offered in Section IV. In Section V cosmological parameters are covered. In Section VI, a few models of "f(Q,T)" gravity are discussed. We have taken a function $f(Q,T)$ both linear and quadratic as $f(Q,T)={a}_{1}Q+{a}_{2}T$ and $f(Q,T)={a}_{3}{Q}^{2}+{a}_{4}T$ where ${a}_{1}$, ${a}_{2}$, ${a}_{3}$ and ${a}_{4}$ are model parameters. A summary of concluding remarks is provided in the last Section VII.

The modified Einstein-Hilbert action principle for the $f(Q,T)$ extended symmetric teleparallel gravity is given by [26]
where $f(Q,T)$ being the general functional form of the non-metricity scalar Q and the trace of the energy-momentum tensor T. g is the determinant of the metric tensor ${g}_{\mu \nu}$ i.e. $g=det\left(\right)open="("\; close=")">{g}_{\mu \nu}$, and ${L}_{m}$ is Lagrangian matter. The non-metricity scalar Q is defined as
$$Q\equiv -{g}^{\mu \nu}\left(\right)open="("\; close=")">{L}^{\delta}{}_{\alpha \mu}{L}^{\alpha}{}_{\nu \delta}-{L}^{\delta}{}_{\alpha \delta}{L}^{\alpha}{}_{\mu \nu}$$
where the deformation tensor ${L}^{\delta}{}_{\alpha \gamma}$ is given by
$${L}^{\delta}{}_{\alpha \gamma}=-\frac{1}{2}{g}^{\delta \eta}\left(\right)open="("\; close=")">{\nabla}_{\gamma}{g}_{\alpha \eta}+{\nabla}_{\alpha}{g}_{\eta \gamma}-{\nabla}_{\eta}{g}_{\alpha \gamma}$$

$$S=\int \left(\right)open="["\; close="]">\frac{1}{16\pi}f(Q,T)+{L}_{m}$$

The non-metricity tensor has the following definition:
and the non-metricity tensor’s trace is derived as
$${Q}_{\delta}={g}^{\mu \nu}{Q}_{\delta \mu \nu},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}{\tilde{Q}}_{\delta}={g}^{\mu \nu}{Q}_{\mu \delta \nu}.$$

$${Q}_{\gamma \mu \nu}={\nabla}_{\gamma}{g}_{\mu \nu},$$

Further, define the super potential tensor as follows
$${P}_{\mu \nu}^{\delta}=-\frac{1}{2}{Q}_{\mu \nu}^{\delta}+\frac{1}{4}\left(\right)open="("\; close=")">{Q}^{\delta}-{\tilde{Q}}^{\delta}$$
using this the non-metricity scalar is

$$Q=-{Q}_{\delta \mu \nu}{P}^{\delta \mu \nu}.$$

The variation of the energy-momentum tensor with respect to the metric tensor ${g}_{\mu \nu}$ read as
$$\frac{\delta \left({g}^{\mu \nu}{T}_{\mu \nu}\right)}{\delta {g}^{\alpha \beta}}={T}_{\alpha \beta}+{\theta}_{\alpha \beta}.$$
where the energy-momentum tensor is
$${T}_{\mu \nu}=\frac{-2}{\sqrt{-g}}\frac{\delta \left(\right)open="("\; close=")">\sqrt{-g}{L}_{m}}{}\delta {g}^{\mu \nu}$$
and

$${\theta}_{\mu \nu}={g}^{\alpha \beta}\frac{\delta {T}_{\alpha \beta}}{\delta {g}^{\mu \nu}}.$$

Also, the field equations of $f\left(\right)open="("\; close=")">Q,T$ gravity are given by varying the action, Equation (1), with respect to the metric tensor ${g}_{\mu \nu}$,
$$-\frac{2}{\sqrt{-g}}{\nabla}_{\delta}\left(\right)open="("\; close=")">{f}_{Q}\sqrt{-g}{P}^{\delta}{}_{\mu \nu}-{f}_{Q}\left(\right)open="("\; close=")">{P}_{\mu \delta \alpha}{Q}_{\nu}{}^{\delta \alpha}-2{Q}^{\delta \alpha}{}_{\mu}{P}_{\delta \alpha \nu}$$
where ${f}_{Q}=\frac{df\left(\right)open="("\; close=")">Q,T}{}dQ$, ${f}_{T}=\frac{df\left(\right)open="("\; close=")">Q,T}{}dT$, and ${\nabla}_{\delta}$ denotes the covariant derivative. From Equation (11) it emerge that the field equations of $f\left(\right)open="("\; close=")">Q,T$ depend on the tensor ${\theta}_{\mu \nu}$. Various cosmological models of $f\left(\right)open="("\; close=")">Q,T$ gravity, depending on the nature of the source of matter are possible. Very recently Koussour et al. [43] have investigated the quintessence form of extended symmetric teleparallel gravity with cosmic acceleration by assuming the cosmic time-redshift relation as $t\left(z\right)=\frac{n{t}_{0}}{m}g\left(z\right)$ which gives the Hubble parameter and verify the sustainability of the results through the energy conditions along with jerk parameter. Also, in [44] the authors investigated energy conditions in $f(Q,T)$ gravity for two different forms of models.

In order to find a solution for the field equations in $f(Q,T)$ extended symmetric teleparallel gravity, some straightforward assumptions, such as the selection of a metric, are frequently required. Consequently, we consider the form’s flat FLRW metric,
$$d{s}^{2}=-d{t}^{2}+{a}^{2}\left(t\right)\left(\right)open="("\; close=")">d{x}^{2}+d{y}^{2}+d{z}^{2}$$
where $a\left(t\right)$ is the scale factor of the metric and its depends on cosmic time (where unit of the cosmic time is Gyr). Energy-momentum tensor for the Universe, imagine to be behaves like a perfect fluid is given by

$${T}_{\nu}^{\mu}=diag\left(\right)open="("\; close=")">-\rho ,p,p,p$$

Here, p denotes pressure and $\rho $ denotes energy density of the Universe. Thus, for tensor ${\theta}_{\nu}^{\mu}$, the expression is obtained as ${\theta}_{\nu}^{\mu}=diag\left(\right)open="("\; close=")">2\rho +p,-p,-p,-p$. The Einstein field equations using the metric (12) are given as,
$${\kappa}^{2}\rho =\frac{f}{2}-6F{H}^{2}-\frac{2\tilde{G}}{1+\tilde{G}}\left(\right)open="("\; close=")">\stackrel{.}{F}H+F\stackrel{.}{H}$$
$${\kappa}^{2}p=-\frac{f}{2}+6F{H}^{2}+2\left(\right)open="("\; close=")">\stackrel{.}{F}H+F\stackrel{.}{H}$$

Hence overhead $\left(\text{\xb7}\right)$ represents a derivative with respect to cosmic time $\left(t\right)$. In this case, $F\equiv {f}_{Q}$ and ${\kappa}^{2}\tilde{G}\equiv {f}_{T}$ represent differentiation of function $f(Q,T)$ with respect to Q and T respectively and $Q=6{H}^{2}$ . With the help of Equations (14) and (15), the EoS parameter is expressed as
$$\omega =-1+\left(\right)open="("\; close=")">\frac{1}{{\kappa}^{2}\rho}\left(\right)open="("\; close=")">\stackrel{.}{F}H+F\stackrel{.}{H}$$

Now we provide a glimpse on the main features of the scale factor (transit scale factor) and derive a few physical quantities from them to discuss the observed scenario. Hence for the transit scale factor, Hubble’s parameter is observed as
where $\u03f5$, $\lambda $ and $\delta $ are the model parameters. Keep in mind the relation of a and z as $a=\frac{1}{1+z}$, Equation (17) becomes
$$H\left(z\right)=\frac{{H}_{0}}{(\lambda +1)}\left(\right)open="("\; close=")">\lambda +{(1+z)}^{\delta}$$
here ${H}_{0}$ represents the present Hubble constant which explains the present expansion rate of the Universe. Freedman et al. [45] and Suyu et al. [46] evaluated a value of the present Hubble constant ${H}_{0}=72\pm 8km/s/Mpc$ and $69.{7}_{-5}^{+4.9}km/s/Mpc$ where as recently Plank gives ${H}_{0}=67.3\pm 1.20km/s/Mpc$ [47].

$$H\left(z\right)=\u03f5\left(\right)open="("\; close=")">{a}^{-\delta}+\lambda $$

Using the time-redshift differential relation $\frac{d}{dt}=-(1+z)H\frac{d}{dz}$, the first and second derivative of H is obtained as
$$\begin{array}{c}\hfill \dot{H}\left(z\right)=-\frac{\delta {H}_{0}^{2}{(z+1)}^{\delta}\left(\right)open="("\; close=")">\lambda +{(z+1)}^{\delta}}{}{(\lambda +1)}^{2}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\end{array}$$

It is interesting to observe and compare our investigated model with the recent observations by finding the best-fit values and the best-fit curve of the Hubble function towards the model parameters ${H}_{0}$, $\u03f5$, $\lambda $, and $\delta $ with the recent observational datasets. Hence, here we used the most favorable Hubble and Supernovae SNe-Ia to constrain the said model parameter in the following section.

The accelerated expansion can be explained by a cosmological constant; alternative explanations, such as dynamical dark energy or modified gravity, can be investigated by looking at how they affect the history of the universe’s late-time expansion or the development of its structures. In order to identify constant parameters, we used the Hubble and Supernova Ia Datasets in this research.

We notice the observational constraints on the parameters ${H}_{0}=\u03f5(\lambda +1)$, $\delta $ and $\lambda $ using the latest 54 data points of $H\left(z\right)$ in the red-shift range $0.07\le z\le 2.4$ in which 28 points obtained using DA method, whereas 26 points using BAO. The values are presented in Table 2.

We have found best-fit curve of $H\left(z\right)$ with 54 data observed values shown in Table 2, using ${R}^{2}-test$:
$${R}^{2}=1-\frac{{\sum}_{1}^{54}{[{\left({H}_{i}\right)}_{obs}-{\left({H}_{i}\right)}_{th}]}^{2}}{{\sum}_{1}^{54}{[{\left({H}_{i}\right)}_{obs}-{\left({H}_{i}\right)}_{mean}]}^{2}}$$

If ${R}^{2}=1$ shows an exact fit the values of model parameters ${H}_{0}$,$\delta $ and $\lambda $ with observational datasets. From Equation (17) we required $-1<z$ and $\u03f5\ne 0$. To find the best-fit values of ${H}_{0}$ ,$\delta $ and $\lambda $ we have restricted the parametric space $-1<z$ and $\u03f5\ne 0$. We used error bars to represent the mean point and its deviation from the mean for 54 points of Hubble data-set and compared our model with the well-accepted $\Lambda $ CDM model for ${H}_{0}=67.8km/s/Mpc$, ${\Omega}_{{\Lambda}_{0}}=0.7$ and ${\Omega}_{{m}_{0}}=0.3$. We have obtained the best-fit plot for approx values ${H}_{0}=64.{49}_{-0.32}^{+0.33}$,$\delta =1.{54}_{-0.02}^{+0.02}$ and $\lambda =1.{14}_{-0.077}^{0.068}$ having maximum ${R}^{2}=0.9321$ with RMSE is 11.071 as shows in Figure 1. Therefore ${H}_{0}=64.{4772}_{-0.32}^{+0.33}km/s/Mpc$ with $6.79\%$ away from exact fit. The Figure 3 show the $1-\sigma $ (dark blue shaded), $2-\sigma $ (Sky blue shaded) maximum likelihood contours in the ${H}_{0}$-$\delta $, ${H}_{0}$-$\lambda $ and $\lambda $-$\delta $ planes for Hubble Datasets.

Distance modulus $\mu \left(z\right)$=$m-M$ given by
where M is constant for all SNe Ia. To obtain the best-fit curve we have to consider data frame having 580 entries observed of apparent magnitude from union 2.1 compilation [48] where ${d}_{L}\left(z\right)=(1+z){\int}_{0}^{z}\frac{{H}_{0}}{H\left({z}^{*}\right)}d{z}^{*}$. From the [48] statistically significant value of M is -19.30.
$${R}^{2}=1-\frac{{\sum}_{1}^{580}{[{\left({\mu}_{i}\right)}_{obs}-{\left({\mu}_{i}\right)}_{th}]}^{2}}{{\sum}_{1}^{580}{[{\left({\mu}_{i}\right)}_{obs}-{\left({\mu}_{i}\right)}_{mean}]}^{2}}$$

$$\mu \left(z\right)=5{log}_{10}{d}_{L}-5{log}_{10}\left(\frac{{H}_{0}}{Mpc}\right)+25$$

If ${R}^{2}=1$ shows exact fit the values of model parameters ${H}_{0}$ ,$\delta $ and $\lambda $ with observational datasets. We used error bars to represent the mean point and its deviation from the mean for 580 points of SNe-Ia Dataset and compared our model with the well-accepted $\Lambda $ CDM model for ${H}_{0}=67.8km/s/Mpc$, ${\Omega}_{{\Lambda}_{0}}=0.7$ and ${\Omega}_{{m}_{0}}=0.3$. We have obtained the best-fit plot for approx values ${H}_{0}=68.{665}_{-2.1}^{+2.2}$,$\delta =1.{53}_{-0.29}^{+0.28}$ and $\lambda =1.{86}_{-0.34}^{+0.37}$ having maximum ${R}^{2}=0.9930$ with RMSE is 0.2662 as shows in Figure 2.

Therefore ${H}_{0}=68.{665}_{-2.1}^{+2.2}km/s/mpc$ with $0.7\%$ away from exact fit. The Figure 3 show the $1-\sigma $ (dark green shaded), $2-\sigma $ (Light green shaded) maximum likelihood contours in the ${H}_{0}$-$\delta $, ${H}_{0}$-$\lambda $ and $\lambda $-$\delta $ planes for SNe-Ia Datasets.

Immediately, we accepted a set of solutions for the planned plan. In order to talk about how Universe is evolved in various phases, we need discuss the behavior of some cosmological parameters like the deceleration parameter, statefinder parameter etc., and these are defined as:

**The deceleration parameter $\left(\mathit{q}\right)$ is**

$$q=-1+\frac{d}{dt}\left(\right)open="("\; close=")">\frac{1}{H}$$

Here the decelerating phase refers to $q>0$ while $q<0$ corresponds to the accelerating phase of the universe. for $q=0$ is the transition point for the Universe from the deceleration to the acceleration phase.

**The statefinder parameters are**

The flat $\Lambda $CDM model is shown at the point where the statefinder parameters $\left(\right)open="\{"\; close="\}">r,\mathrm{s}$ have the corresponding values. Additionally, keep in mind that in the $\left(\right)$ plane, a positive parameter s (i.e., $s>0$) denotes a quintessence-like model of dark energy, whereas a negative parameter s (i.e., $s<0$) denotes a phantom-like model of dark energy. Furthermore, by traversing the point $\left(\right)open="\{"\; close="\}">r,\mathrm{s}$, one can evolve from a phantom to the quintessence.

**Energy conditions**

One can derive highly potent and broad theorems regarding the behavior of massive gravitational fields and cosmic geometries using the energy conditions (ECs) of general relativity (GR). Generally speaking, ECs can be divided into

- Strong Energy Condition (SEC): Gravity should always be attractive and it formulated as $\rho +3p\ge 0$.
- Dominant Energy Condition (DEC): When observer measures a matter energy density then will be positive and propagate in a causal way, which leads to $\rho \ge \left|p\right|$.
- Weak Energy Condition (WEC): The matter energy density measured by any observer should be positive, $\rho \ge 0,\rho +p\ge 0$.
- Null Energy Condition (NEC): It’s the minimum requirement that is implied by SEC and WEC, is $\rho +p\ge 0$.

The violation of the NEC in the energy conditions implies that none of the energy criteria given are valid. The current fast expansion of the universe has raised questions about the SEC. In cosmological situations during the inflationary expansion and at the present, SEC must be broken.

Making use of Equations (18) and (19), the deceleration parameter and statefinder parameters are obtained as
$$r\left(z\right)=\frac{{\lambda}^{2}+(\delta -2)(\delta -1)\lambda {(z+1)}^{\delta}+(\delta -1)(2\delta -1){(z+1)}^{2\delta}}{{\left(\right)}^{\lambda}}$$
$$s\left(z\right)=1+\frac{\delta {(z+1)}^{\delta -1}\left(\right)open="("\; close=")">(z+1)\u03f5\left(\right)open="("\; close=")">(2\delta -3){(z+1)}^{\delta}-3\lambda}{\left(\right)}-\delta \lambda \u03f5{\left(\right)}^{\lambda}3$$

$$q=-1+\frac{\delta {(z+1)}^{\delta}}{\lambda +{(z+1)}^{\delta}}.$$

After analysing the SNe-Ia data by many researchers it was observed that datasets favour current acceleration for $(z<0.5)$ and past deceleration for $(z>0.5)$. A little while back, according to the high-z supernova search (HZSNS) team ${z}_{0}=0.46\pm 0.130$ at (1$\sigma $ ) confidence level [49] which has been further analyzed to ${z}_{0}=0.43\pm 0.070$ at (1 $\sigma $ ) [49]. According to SNLS [50], as well as the one recently compiled in [51], the transition red-shift ${z}_{0}$≡ 0.6 (1$\sigma $) is in better agreement with the flat $\Lambda $CDM model ${z}_{0}={(2{\Omega}_{\Lambda}/{\Omega}_{m})}^{1/3}-1\sim 0.66)$. Another limit is $0.60\le {z}_{0}\le 1.18$ (2$\sigma $, joint analysis) [52]. Further, the transition red-shift for our derived model comes to be ${z}_{0}\cong 0.65$ for observed Hubble datasets and ${z}_{0}\cong 1.965$ for supernovae which is in best agreement with the SNe- Ia supernovae observations, including the farthest known supernova SNI997ff at $z\approx 1.7$ [53]. We see that the variation of q with z obtained in our model is compatible with the results. In our derived model the best-fit value of deceleration parameter ${q}_{0}$ for Hubble and Supernova is -0.2792 , -0.4774. Figure 4 shows plot of deceleration parameter v Redshift for both Hubble and Supernovae datasets for values of model parameters ${H}_{0}$, $\delta $ and $\lambda $ are from Table 1.

In this study, it was argued that $\left(\right)$ plane is useful to differentiate between various models. An analysis based on $\left(\right)$ has also useful to differentiate between general relativity and modified theory of gravity. We note that for the Hubble datasets, the r and s parameters at the present epoch are ${r}_{0}=0.43987$ and ${s}_{0}=-3.3632$ while for the SNe-Ia datasets, ${r}_{0}=0.48918$ and ${s}_{0}=-10.5851$ as shown in Figure 5. Currently, observations are not sensitive enough to measure these parameters. Future data, however, could be used to infer these values, which would tremendously aid in defining the characteristics of dark energy.

In this section, we discuss some physical aspects of different models of $f(Q,T)$ gravity

Here we consider the $f(Q,T)$ gravity model as

$$f(Q,T)={a}_{1}Q+{a}_{2}T,$$

For above model the field Equations (14)–(16), take the form
$$\rho =\frac{{a}_{1}\dot{H}}{{a}_{2}+8\pi}-\frac{{a}_{1}\left(\right)open="("\; close=")">3{H}^{2}+\dot{H}}{}2\left(\right)open="("\; close=")">{a}_{2}+4\pi ,$$

Figure 6 shows a plot of Pressure versus Redshift for both Hubble and SNe-Ia datasets for ${H}_{0}$, $\delta $ and $\lambda $ are from Table 1 while model parameters ${a}_{1}$=-0.0125 and ${a}_{2}$=-0.012 respectively.

$$\omega =\frac{-3{a}_{2}\left(\right)open="("\; close=")">{H}^{2}+\dot{H}}{-}$$

The EoS parameter is associated with energy density $\rho $ and pressure p. The EoS parameter appears to be positive in the beginning. As a consequence, it moves from the positive region to negative region. The negative $\omega $ is proposed a constant vacuum energy density, It’s worth noting that $\omega =0$ shows Pressure-less Cold matter (PCL), $\omega =(0,\frac{1}{3})$ represents Hot matter, $\omega =\frac{1}{3}$ is radiation, $\omega =(\frac{1}{3},1)$ is Hard Universe, $\omega =1$ shows stiff fluid (SF), $\omega >1$ is Ekpyrotic matter (Ek-M), $\omega >-1$ stand for the quintessence (Q) region and $\omega <-1$ stands for the phantom region (Ph) , respectively while $\omega =-1$ represents the cosmological constant ($\Lambda $ CDM) and $\omega <<-1$ is precluded by SNe-Ia perceptions. Subsequently, the evolving range of $\omega $ of our derived model is supportive of ($\Lambda $ CDM) model in both Hubble and supernova data.

From Figure 7, we can observe that the Universe exists the decelerated regime and enters in the accelerating phase as studied [54].

Furthermore, to verify the genuineness of model in context of cosmic acceleration, we resolve different forms of energy conditions by calculating
and
$$3p+\rho ={a}_{1}\left(\right)open="("\; close=")">\frac{3{H}^{2}+\dot{H}}{{a}_{2}+4\pi}+\frac{4\dot{H}}{{a}_{2}+8\pi}$$
for NEC, DEC and SEC respectively. Figure 8 and Figure 9 are plots of Energy Conditions with respect to constant obtained from best-fit for Hubble and SNe-Ia datasets as shown in Table 1 and model parameters ${a}_{1}$=-0.0125 and ${a}_{2}$=-0.012. According to the both data of the accelerating Universe, the SEC must be violated on cosmological scale [55,56]. Also, negative EoS ( $\omega $) indicate that $\rho +3p<0$. Therefore, there is a violation of the SEC at present. We also can see in Figure 8 and Figure 9 that the NEC, and DEC are obeying for both Hubble and SNe-Ia datasets. Since we have shown the behavior of energy density in Figure 6. We have examined the NEC behavior (i.e., partial condition of WEC). Therefore, validation of NEC and energy density together results in the validation of WEC.

$$\rho +p=\frac{2{a}_{1}\dot{H}}{{a}_{2}+8\pi},$$

$$\rho -p=-\frac{{a}_{1}\left(\right)open="("\; close=")">3{H}^{2}+\dot{H}}{}{a}_{2}+4\pi $$

z | H(z) | ${\mathit{\sigma}}_{\mathit{H}}$ | Ref. | z | H(z) | ${\mathit{\sigma}}_{\mathit{H}}$ | Ref. |
---|---|---|---|---|---|---|---|

0.07 | 69 | 19.6 | [57] | 0.9 | 69 | 12 | [58] |

0.120 | 68.6 | 26.2 | [57] | 0.170 | 83 | 8 | [59] |

0.179 | 75 | 4 | [60] | 0.2 | 72.9 | 29.6 | [57] |

0.27 | 77 | 14 | [59] | 0.28 | 88.8 | 36.6 | [57] |

0.350 | 76.3 | 5.6 | [61] | 0.38 | 83 | 13.5 | [62] |

0.4 | 95 | 17 | [59] | 0.42 | 87.1 | 11.2 | [62] |

0.44 | 92.8 | 12.9 | [62] | 0.47 | 89 | 34 | [57] |

0.48 | 97 | 62 | [63] | 0.6 | 87.9 | 6.1 | [64] |

0.68 | 92 | 8 | [60] | 0.73 | 97.3 | 7 | [64] |

0.78 | 105 | 12 | [60] | 0.87 | 125 | 17 | [60] |

0.90 | 117 | 23 | [59] | 1.037 | 154 | 20 | [60] |

1.3 | 168 | 17 | [59] | 1.363 | 160 | 33.6 | [59] |

1.430 | 177 | 18 | [59] | 1.530 | 140 | 14 | [59] |

1.750 | 202 | 40 | [59] | 1.965 | 186.5 | 50.4 | [65] |

0.24 | 79.69 | 2.99 | [66] | 0.30 | 81.7 | 6.22 | [67] |

0.31 | 78.18 | 4.74 | [68] | 0.34 | 83.8 | 3.66 | [66] |

0.35 | 87.7 | 9.1 | [69] | 0.36 | 79.94 | 3.38 | [68] |

0.38 | 81.5 | 1.9 | [70] | 0.40 | 82.04 | 2.03 | [68] |

0.43 | 86.45 | 3.97 | [66] | 0.44 | 82.6 | 7.8 | [71] |

0.44 | 84.81 | 1.83 | [68] | 0.48 | 87.79 | 2.03 | [68] |

0.51 | 90.4 | 1.9 | [70] | 0.52 | 94.35 | 2.64 | [68] |

0.56 | 93.34 | 2.3 | [68] | 0.57 | 87.6 | 7.8 | [72] |

0.57 | 96.8 | 3.4 | [73] | 0.59 | 98.48 | 3.18 | [68] |

0.6 | 87.9 | 6.1 | [71] | 0.61 | 97.3 | 2.1 | [70] |

0.64 | 98.82 | 2.98 | [68] | 0.73 | 97.3 | 7 | [71] |

2.30 | 224 | 8.6 | [74] | 2.33 | 224 | 8 | [75] |

2.34 | 222 | 8.5 | [76] | 2.36 | 226 | 9.3 | [77] |

Here we consider the $f(Q,T)$ gravity model as,

$$f(Q,T)={a}_{3}{Q}^{2}+{a}_{4}T,$$

For above model the field Equations (14)–(16), take the form
$$\rho =-\frac{3\left(\right)open="("\; close=")">72\pi {a}_{3}{H}^{4}+9{a}_{3}{a}_{4}{H}^{4}-2{a}_{3}{a}_{4}{H}^{2}\dot{H}-4{a}_{3}{a}_{4}\dot{H}{H}^{2}}{}\left(\right)open="("\; close=")">{a}_{4}+4\pi \left(\right)open="("\; close=")">{a}_{4}+8\pi $$

Figure 10 shows plot of Density versus redshift for both Hubble and SNe-Ia datasets for ${H}_{0}$, $\delta $ and $\lambda $ are from Table 1, while model parameters ${a}_{1}$=-0.0125 and ${a}_{2}$=-0.012 respectively.

$$\omega =\frac{-3{a}_{4}\left(\right)open="("\; close=")">3{H}^{2}+4\dot{H}+2\dot{H}}{-}$$

As shown in Figure 11, the universe is in an accelerating mode and is about to enter a decelerating phase. Additionally, we resolve various energy conditions by calculating the model’s accuracy in the context of cosmic acceleration.
$$\rho -p=-\frac{6{a}_{3}{H}^{2}\left(\right)open="("\; close=")">9{H}^{2}+4\dot{H}+2\dot{H}}{}{a}_{4}+4\pi $$
and
$$3p+\rho =\frac{6{a}_{3}{H}^{2}\left(\right)open="("\; close=")">{a}_{4}\left(\right)open="("\; close=")">9{H}^{2}+20\dot{H}+10\dot{H}}{+}\left(\right)open="("\; close=")">{a}_{4}+4\pi \left(\right)open="("\; close=")">{a}_{4}+8\pi $$
for NEC, DEC and SEC, respectively.

$$\rho +p=\frac{24{a}_{3}{H}^{2}\left(\right)open="("\; close=")">2\dot{H}+\dot{H}}{}{a}_{4}+8\pi $$

For NEC, DEC and SEC respectively. Figure 12 and Figure 13 are plots of Energy Conditions with respect to constant obtained from best-fit for Hubble and SNe-Ia datasets as shown in Table 1 and model parameters ${a}_{1}$=-0.0125 and ${a}_{2}$=-0.012. According to the both data of the accelerating Universe,as shown in Figure 12 and Figure 13 the SEC must be violated on cosmological scale [55,56]. Also, negative EoS ( $\omega $) indicate that $\rho +3p<0$. Therefore, there is a violation of the SEC at present. Figure 12 and 13 indicate that the NEC, and DEC are obeying for both Hubble and SNe-Ia datasets. We have examined the NEC behavior (i.e., partial condition of WEC). Therefore, validation of NEC and energy density together results in the validation of WEC.

In present work we have taken a function $f(Q,T)$ both linear and quadratic as
where ${a}_{1}$, ${a}_{2}$, ${a}_{3}$ and ${a}_{4}$ are model parameters. In terms of redshift z, we have measured a number of cosmological parameters in the FLRW universe, including the Hubble parameter H and the deceleration parameter q. By applying the ${R}^{2}-test$ formula for observational constraints on the model, we were able to determine the approximate best-fit values of the model parameters $\u03f5,\delta ,\lambda ,\mathrm{and}{H}_{0}$ utilizing datasets like the Hubble data set $H\left(z\right)$ and union 2.1 compilation of SNe-Ia datasets. The current values of the cosmological models ${H}_{0}$ and ${q}_{0}$ that we estimated are close to the values found in mainstream cosmology. Talk about EoS and the various energy conditions. Following are the characteristics of our cosmological model:

$$f(Q,T)={a}_{1}Q+{a}_{2}T,f(Q,T)={a}_{3}{Q}^{2}+{a}_{4}T$$

- Figure 1 and Figure 2 provide best-fit plots based on the observed datasets. For the best fit, we employed a hybrid model combining the gradient descent and least squares approach. ${R}^{2}$-value for Hubble and SNe-Ia datasets are $0.9321$ and $0.9930$ respectively. SNe-Ia has 580 observation and giving best-fit among both datasets.
- Hubble function derived is constrained by observational datasets i.e. Hubble and SNe-Ia datasets and the present value of Hubble constant is $64.4772km/s/Mpc$ and $68.665km/s/Mpc$ respectively with respect to best-fit plot.
- The transition from early deceleration to the universe’s present acceleration is shown by the deceleration parameter ${q}_{0}=-0.2792$ and ${q}_{0}=-0.4774$ with respect to Hubble and SNe-Ia datasets.
- We have considered two functional form of $f(Q,T)$ gravity in Section A, Section B and watched how the energy density and EoS parameter all behaved. The EoS parameter exhibits a transition from early deceleration to late time acceleration with regard to the model parameters, while the density in both models exhibits positive behavior.

No advanced data associated with this article.

The IUCAA, Pune, India, provided the facilities and support under the Visiting Associateship Programmes, for which the authors (S. H. Shekh and A. Pradhan) are grateful.

The authors declared that they have neither personal relationships nor competing financial interests that could influenced the work reported in this paper.

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Datasets | ${\mathit{H}}_{0}$ | $\mathit{\delta}$ | $\mathit{\lambda}$ | $\mathit{\u03f5}$ |
---|---|---|---|---|

Hubble | $64.{49}_{-0.32}^{+0.33}$ | $1.{54}_{-0.02}^{+0.02}$ | $1.{14}_{-0.077}^{+0.068}$ | $30.{2}_{-0.87}^{+0.90}$ |

SNe-Ia | $68.{665}_{-2.1}^{+2.2}$ | $1.{53}_{-0.29}^{+0.28}$ | $1.{86}_{-0.34}^{+0.37}$ | $23.{954}_{-2.84}^{+3.74}$ |

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